论文标题
与服务和耐心时间完全相关的排队系统的多个服务器繁重的限制
Many-Server Heavy-Traffic Limits for Queueing Systems with Perfectly Correlated Service and Patience Times
论文作者
论文摘要
当客户的服务要求与他们的个人耐心等待排队时，我们对根据Square-Root安全规则配备的系统进行了繁重的流量过程和稳态限制。在通常的多个服务器扩散缩放下，我们表明该系统在不放弃的情况下渐近地等同于系统。特别是，当交通强度接近其关键值$ 1 $时，限制是$ m/m/n $队列的半旋流扩散，尽管预序是正面的，但否则是短暂的扩散。为了获得由于相关性引起的拥塞的精致度量，我们表征了瞬时扩散极限的情况下的低阶流体（LOF）限制，这表明在这种情况下，队列在这种情况下的尺度如$ n^{3/4} $。在扩散和LOF量表下，我们表明固定分布薄弱地收敛到相应过程限制的时间限制行为。
We characterize heavy-traffic process and steady-state limits for systems staffed according to the square-root safety rule, when the service requirements of the customers are perfectly correlated with their individual patience for waiting in queue. Under the usual many-server diffusion scaling, we show that the system is asymptotically equivalent to a system with no abandonment. In particular, the limit is the Halfin-Whitt diffusion for the $M/M/n$ queue when the traffic intensity approaches its critical value $1$ from below, and is otherwise a transient diffusion, despite the fact that the prelimit is positive recurrent. To obtain a refined measure of the congestion due to the correlation, we characterize a lower-order fluid (LOF) limit for the case in which the diffusion limit is transient, demonstrating that the queue in this case scales like $n^{3/4}$. Under both the diffusion and LOF scalings, we show that the stationary distributions converge weakly to the time-limiting behavior of the corresponding process limit.
